Theorem sbcieg 3814

Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 10-Nov-2005.) Avoid ax-10 2129, ax-12 2166. (Revised by GG, 12-Oct-2024.)

Hypothesis

Ref	Expression
sbcieg.1	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
sbcieg	⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))

Distinct variable groups:   𝑥,𝐴   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem sbcieg

Step	Hyp	Ref	Expression
1		df-sbc 3774	. 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})
2		sbcieg.1	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
3	2	elabg 3662	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))
4	1, 3	bitrid 282	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1533   ∈ wcel 2098  {cab 2702  [wsbc 3773

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-sbc 3774

This theorem is referenced by:  sbcie  3817  2nreu  4443  reuprg0  4708  rabsnif  4729  ralrnmptw  7103  ralrnmpt  7105  fpwwe2lem3  10658  nn1suc  12267  opfi1uzind  14498  mndind  18788  fgcl  23826  cfinfil  23841  csdfil  23842  supfil  23843  fin1aufil  23880  ifeqeqx  32412  nn0min  32668  bnj1452  34811  cdlemk35s  40537  cdlemk39s  40539  cdlemk42  40541  2nn0ind  42505  zindbi  42506  prproropreud  46983